Differentiable and Complex Dynamics of Several Variables

by Hu, Pei-Chu; Yang, Chung-Chun

• ISBN13: 9780792357711
• ISBN10: 079235771X
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Kluwer Academic Pub

Summary

This book gives a comprehensive and up-to-date survey on dynamics and related topics, such as Fatou-Julia type theory, the Ergodic theorem and invariant sets, hyperbolicity in differentiable or complex dynamics, iterant ion theory on P m , complex dynamics in C m and the foundations of differentiable and complex dynamics. The main aims of this volume are, firstly, to advance the study of the above-named topics and to establish the corresponding Fatou-Julia results for complex manifolds, and, secondly, to provide some advanced account of dynamical systems within the framework of geometry and analysis, presented from a unified approach applicable to both real and complex manifolds. Audience: This work will be of interest to graduate students and researchers involved in the fields of global analysis, analysis on manifolds, several complex variables and analytic spaces, partial differential equations, differential geometry, measure and integration.

Table of Contents

Preface	vii
Fatou-Julia type theory	1	(38)
Hyperbolic fixed points	1	(3)
Attractive and repulsive fixed points	4	(5)
Normality	9	(8)
Equicontinuity	17	(3)
Lipschitz constants and Julia sets	20	(6)
Degree of mappings and Julia sets	26	(6)
Topological entropies and Julia sets	32	(7)
Ergodic theorems and invariant sets	39	(24)
Poincare recurrence domains	39	(4)
Neumann domains	43	(4)
Birkhoff domains	47	(3)
Ergodic points	50	(4)
Regular ergodic points	54	(4)
Lp-Ergodic points	58	(2)
Lo-Ergodic points	60	(3)
Hyperbolicity in differentiable dynamics	63	(36)
Nonwandering sets and chain recurrent sets	63	(5)
Measure theoretic entropies	68	(4)
Lyapunov exponents	72	(6)
Hyperbolic sets	78	(6)
Notes on hyperbolic sets	84	(5)
Mappings with bounded distortion	89	(4)
Hausdorff measure and metric	93	(6)
Some topics in dynamics	99	(38)
Properties of Hamiltonian systems	99	(6)
Hamiltonian systems near an equilibrium solution	105	(4)
Linearization of equations	109	(5)
Linearization of mappings	114	(8)
Lp-Normality	122	(5)
Lp-Equicontinuity	127	(2)
Cr-hyperbolicity	129	(8)
Hyperbolicity in complex dynamics	137	(42)
Complex dynamics	137	(6)
Schwarz lemma	143	(8)
Kobayashi hyperbolicity	151	(7)
Tautness and tightness	158	(4)
Relations between hyperbolicity and tautness	162	(4)
Julia sets of meromorphic mappings	166	(6)
A generalization of Marty's criterion	172	(7)
Iteration theory on Pm	179	(24)
Meromorphic self-mappings on Pm	179	(5)
Fatou sets in Pm	184	(3)
Periodic points	187	(3)
Classification of recurrent domains on P2	190	(5)
Exceptional varieties and critical sets	195	(2)
The Newton's method	197	(6)
Complex dynamics in Cm	203	(30)
Iteration theory on domains of Cm	203	(5)
Holomorphic self-mappings on Cm	208	(5)
The group Aut (C2)	213	(5)
Dynamics in C2	218	(5)
Borel's theorem on Cm	223	(5)
Dynamics of composite mappings	228	(5)
A Foundations of differentiable dynamics	233	(42)
A.1 Basic notations in differentiable geometry	233	(9)
A.2 Dynamical systems	242	(6)
A.3 Fixed points and cycles	248	(6)
A.4 Orbits and invariant sets	254	(5)
A.5 The isometric R-actions	259	(4)
A.6 Stability	263	(5)
A.7 Measure and integral	268	(7)
Foundations of complex dynamics	275	(44)
B.1 Complex structures	275	(6)
B.2 Hermitian structures	281	(7)
B.3 Complex manifolds	288	(8)
B.4 Hamiltonian manifolds	296	(7)
B.5 Complex Euclidean spaces Cm	303	(6)
B.6 Complex projective spaces Pm and torus Cm/Γ	309	(3)
B.7 Meromorphic mappings and order functions	312	(7)
Bibliography	319	(16)
Index	335